Canceling branch points on projections of surfaces in -space

Authors:
J. Scott Carter and Masahico Saito

Journal:
Proc. Amer. Math. Soc. **116** (1992), 229-237

MSC:
Primary 57Q35; Secondary 57Q45

MathSciNet review:
1126191

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Abstract: A surface embedded in -space projects to a generic map in -space that may have branch points--each contributing to the normal Euler class of the surface. The sign depends on crossing information near the branch point. A pair of oppositely signed branch points are geometrically canceled by an isotopy of the surface in -space. In particular, any orientable manifold is isotopic to one that projects without branch points. This last result was originally obtained by Giller. Our methods apply to give a proof of Whitney's theorem.

**[1]**Thomas F. Banchoff,*Double tangency theorems for pairs of submanifolds*, Geometry Symposium, Utrecht 1980 (Utrecht, 1980) Lecture Notes in Math., vol. 894, Springer, Berlin-New York, 1981, pp. 26–48. MR**655418****[2]**Thomas F. Banchoff,*Normal curvatures and Euler classes for polyhedral surfaces in 4-space*, Proc. Amer. Math. Soc.**92**(1984), no. 4, 593–596. MR**760950**, 10.1090/S0002-9939-1984-0760950-4**[3]**-,*The normal Euler class of a polyhedral surface in**-space*, unpublished notes and letters.**[4]**T. F. Banchoff and Ockle Johnson (to appear).**[5]**Sylvain E. Cappell and Julius L. Shaneson,*An introduction to embeddings, immersions and singularities in codimension two*, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 129–149. MR**520529****[6]**Cole A. Giller,*Towards a classical knot theory for surfaces in 𝑅⁴*, Illinois J. Math.**26**(1982), no. 4, 591–631. MR**674227****[7]**Seiichi Kamada,*Nonorientable surfaces in 4-space*, Osaka J. Math.**26**(1989), no. 2, 367–385. MR**1017592****[8]**Dennis Roseman,*Projections of codimension two embeddings*, Knots in Hellas ’98 (Delphi), Ser. Knots Everything, vol. 24, World Sci. Publ., River Edge, NJ, 2000, pp. 380–410. MR**1865719**, 10.1142/9789812792679_0024**[9]**-*Reidemeister-type moves for surfaces in four dimensional space*, preprint.**[10]**Bruce Trace,*A general position theorem for surfaces in Euclidean 4-space*, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 123–137. MR**813108**, 10.1090/conm/044/813108**[11]**Hassler Whitney,*On the topology of differentiable manifolds*, Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., 1941, pp. 101–141. MR**0005300**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1126191-0

Keywords:
Embedded surfaces in -space,
projections,
branch points,
normal Euler number

Article copyright:
© Copyright 1992
American Mathematical Society